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Some properties of multivariate extreme value distributions and multivariate tail equivalence

  • Rinya Takahashi
Article

Summary

Denote byH ak-dimensional extreme value distribution with marginal distributionHi(x)=Λ(x)=exp(−ex),xR1. Then it is proved thatH(x)=Λ(x1)...Λ(xk) for anyx=(x1, ...,xk) ∈Rk, if and only if the equation holds forx=(0,...,0). Next some multivariate extensions of the results by Resnick (1971,J. Appl. Probab.,8, 136–156) on tail equivalence and asymptotic distributions of extremes are established.

Key words and phrases

Multivariate extreme value distribution multivariate tail equivalence 

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References

  1. [1]
    Billingsley, P. (1968).Convergence of Probability Measures, Wiley, New York.zbMATHGoogle Scholar
  2. [2]
    de Haan, L. (1970).On Regular Variation and Its Application to the Weak Convergence of Sample Extremes, Mathematical Centre Tracts, Vol. 32, Amsterdam.Google Scholar
  3. [3]
    Galambos, J. (1978).The Asymptotic Theory of Extreme Order Statistics, Wiley, New York.zbMATHGoogle Scholar
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    Marshall, A. W. and Olkin, I. (1983). Domains of attraction of multivariate extreme value distributions,Ann. Probab.,11, 168–177.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Resnick, S. I. (1971). Tail equivalence and its applications,J. Appl. Probab.,8, 136–156.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1987

Authors and Affiliations

  • Rinya Takahashi
    • 1
  1. 1.Kobe University of Mercantile MarineKobeJapan

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